NASA Technical Paper 3362 CECOM Technical Report 93-E-2 1993 National Aeronautics and Space Administration Office of Management Scientific and Technical Information Program Analysis of Microstrip Patc, h Antennas With Nonzero Surface Resistance David G. Shively Joint Research Program Office Electronics Integration Directorate Communications Electronics Command Langley Research Center Hampton, Virginia M. C. Bailey Langley Research Center Hampton, Virginia
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NASATechnical
Paper3362
CECOMTechnical
Report93-E-2
1993
National Aeronautics andSpace Administration
Office of Management
Scientific and Technical
Information Program
Analysis of MicrostripPatc, h Antennas WithNonzero SurfaceResistance
David G. Shively
Joint Research Program Office
Electronics Integration Directorate
Communications Electronics Command
Langley Research Center
Hampton, Virginia
M. C. Bailey
Langley Research Center
Hampton, Virginia
Abstract
The scattering pwperties of a microstrip patch antenna with nonzero
surface impedance are examined. The electric field integral equation
for a current element on a grounded dielectric slab is developed fora rectangular geometry by using Galerkin's technique with subdomain
piecewise linear basis functions. The integral equation includes a resistive
boundary condition on the surface of the patch. The incident field on
the patch is expressed as a function of incidence angle. The resulting
system of equations is then solved for the unknown current modes on
the patch, and the radar cross section is calculated for a given scatteringangle. Theoretical results in the form of radar cross section as a function
of frequency are compared with results measured at the NASA LangleyResearch Center.
Symbols
d
_incEtan
scatt an
E_ TL
E_ Ir_
f H_H
U
I Tt] T!
J
Ko
I_ x
/(y
Ki
K2
thickness of dielectric slab
tangential components of incidentelectric field
tangential components of scatteredelectric field
component of electric field
component of electric field
Fourier transform of currentmode mn
dyadic Green's function
amplitude of mode mn
surface current on nficrostrip patchantenna
variables of integration in cylindri-cal coordinates
propagation constant for free space,27r/,Xo
spectral domain transformationvariable for x-direction
spectral domain transforlnationvariable for y-direction
propagation constant for dielectricslab in z-direction
propagation constant for free spacein z-direction
L/r,
Lg
M
ran, pq
N
R
R8
T,_
VTr_7[
(X, y, Z)
(X / , y', Z t)
xm , Yn
Ax
9
Ay
Z
go
dimension of the patch inx-direction
dimension of the patch in
y-direction
number of subdivisions inx-direction
indices specifying the subdomainbasis functions on patch
immber of subdivisions in
y-direction
resistance matrix representingsurface resistance on patch
surface resistance on microstrip
patch antenna
characteristic equation for trans-verse electric modes
characteristic equation for trans-verse magnetic modes
component of excitation voltagevector
coordinates of field point
coordinates of source point
unit vector in x-direction
coordinates of current mode mn
cell size in x-direction
unit vector in y-direction
cell size in y-direction
impedance matrix to be solved
impedance of free space, 377 _
unit vector in z-direction
e,, relative pernfittivity of dielectricslab
0 i, 0 i incident angle of electromagneticVcave
A piecewise linear flmction for current
oil patch
Ao wavelength of electromagnetic field
in free space
Po permeability of free space
1I pulse flmction for the current on
patch
aO0 0-polarized backscatter from
0-polarized incident field
_0o 0-polarized baekscatter from
0-polarized incident field
0oo 0-polarized hackseatter from
0-polarized incident field
cro0 0-polarized backscatter from
0-polarized incident field
a_ radian frequency of electromagneticfield
Introduction
Spectral domain Green's functions, which de-
s('rit)e the electric field radiated by a current sourceon a grounded dMectric slab, were introduced in
the early 1980's. This allowed the development of a
moment method for analyzing perfectly conducting
microstrip patch antennas. This technique accu-rately accounts for dielectric thickness, dielectric
losses, and surface wave losses and can be extended
to include the effects of a cover layer of a different
dielectric constant on top of the antenna. Because
of the spectral nature of the technique, it can easily
be extended to model an infinite array, of patches byexamining only a single unit cell. Also of interest are
the effects of lossy nmterials on the antenna. Lossy
mate.rials on the antenna will decrease the efficiencyof the antenna, and hence the gain of the antenna will
be lowered. This decrease in gain also means thatthe scattering from the antenna will be decreased.
As losses arc added to an antenna, other properties
of the antmma, such ms bandwidth, input impedance,and ra(tiation patterns, will also be altered.
The moment method technique incorporates ei-
ther subdomain or entire domain expansion func-
tions in order to model tile current on the patch.
Bailey and Deshpande (refs. 1 3) have used sub-domain expansion flmctions in order to model rec-
tangular patches. Many other authors (refs. 4 13)have used entire domain expansion functions in or-
der to model rectangular and circular patches. Baileyand Deshpande (ref. 14) have also used entire domain
expansion functions to model an elliptical patch.
The majority of this work has examined the inputimpedance and scattering properties of perfectly con-
ducting patches both as single radiators and as in-
finite arrays. Hansen and Janhsen (ref. 15) have
included a space-varying surface impedance whenmodeling a nficrostrip feed network.
A technique similar to the spectral domain
method uses spatial domain Green's functions with
subdomain expansion functions in order to model
microstrip structures. The disadvantage of this tech-
nique is that it is not easily extended to examine in-
finite arrays. A number of authors (refs. 16 19) have
used this technique to model microstrip patch anten-
nas as single radiators. Mosig (ref. 18) has mentionedthat conductor losses can be included in this model,but no results have been presented.
The boundary condition for the electric field on
a thin resistive sheet has been examined by Senior(refs. 20 23) and is valid as long as the sheet is elec-
trically thin. Using this type of boundary condition,several authors (refs. 24 27) have examined the scat-
tering response of resistive strips and tapered resis-tive strips. This approach has also been used in order
to study frequency selective surfaces (refs. 28 30).The same model for the surface resistance has been
used in the study of superconducting materials andstrip lines (refs. 31 and 32).
This paper will describe spectral domain analysisof imperfectly conducting microstrip patch antentlasby using subdomain basis functions to model the
patch current density. To simplify the analysis, theantenna feed will not be considered. The antenna
is considered to be open circuited from the feed
network, i.e., the feed impedance is infinite. Resultsare presented in the form of radar cross section as a
function of frequency for a few representative casesand are compared with measured results.
Theory
The geometry of a rectangular microstrip patchantenna is shown in figure 1. The patch is assumed
to be electrically thin and located on a groundeddielectric slab of infinite extent. The dielectric slab
has relative permittivity er, relative permeability #r,
Resistivesurface
N 7Ly
where
1 /? /_ Gab (Kx,Kv zl zl)Gab- 47r2 _ _ •
x eJh'J(z-X')e jK'v('q g') dK.r dKy (4)
Perfectconductor
X
2 Current1 modes
1 2. M
, (0 i, O i)
Figure 1. Geometry of a nficrostrip antenna with arbitrary
surface resistance.
and thickness d. The standard ejwt time convention
is assumed. The boundary condition oll the patch is
#TPq mnNote that the -_zx terms will equal zero if
p> (re+l), p< (m-l), or q¢n. Likewise, theRpqm rly_ terms will equal zero if q > (n + 1),q < (n - 1), or p ¢ m. If Rs is constant in the pqsubdomain, equation (32) reduces to
Figure 3. Calculated and measured scattering from a per-feetly conducting rectangular microstrip patch antenna.L_ = 0.75 era; L_j -- 0.75 era; d = 0.07874 cm; er = 2.33;Loss tangent - 0.001; (0 i, 0i) = (60 o, 180o).
response. All the following calculated results havebeen performed with M = N = 12 and F = 5.
Initially, the radar cross section of four microstrippatches, each with a constant resistance profile, was
measured. The computed and measured responses
for a perfectly conducting patch are shown in figure 3.The subdomain result was calculated as described
above and agrees extremely well with the entire do-
main result calculated by J. T. Aberle (Arizona StateUniversity, private communication). The result mea-
sured in the Experimental Test Range (ETR) at theNASA Langley Research Center is slightly shifted infrequency and slightly lower than expected. This is
not totally unexpected and can be attributed to the
physical tolerances of the patch shape, dielectric con-
stant, and dielectric thickness. The rapid fluctua-
tions seen in the measured data are most likely due
to imperfections in the background subtraction per-formed when processing the radar range data. These
subtractions are necessary to approximate the re-
sponse of the patch on an infinitely large groundeddielectric slab. Although the measured and calcu-
lated data do not exhibit as close agreement as is
evident in the subdonmin and entire domain data,
the relative position of the resonant peaks and thescattering levels are fairly close.
The components of the scattered fieht for other
polarizations are shown in figure 4. The (700 responseshows a resonant peak at the same frequency as the(700 response but also contains a peak in the center of
the band where the (700 response does not. The upperresonant peak in the aoo response is not evident in the
(r0O response. The cross-polarized components, (700and c%0 , are both the same aim show all the peaks of
the previous two responses. The current density on
the patch when illunfinated with a 0-polarized plane
wave as in figure 3, is shown in figure 5. Theand _ components of the current are shown at the
first and second resonances of the patch. At the
first resoimnce the current resembles the expectedsinusoidal distribution. At the second resonance this
is also the case, although a whole period of thesinusoid is now evident.
Calculations silnilar to those described above have
been performed for a patch with a surface resistance
of 5 f_ over the entire patch. The calculated and
mea,sured results for (700 are shown in figure 6. Agree-
ment in this case, is not as good as in the previous
7
0-10-20-3O-40-50-60-70
= -80_ -90
-100-1|0-120-130-140-1506
_oo.... o¢,
% .-el __ 7
_s
,F,L s •
• • s SS
_ • I h ,
7 8 9 10 11 12 13 14
Frequency, GHz
Figure 4. Scattering from a perfectly conducting rectangular microstrip patch as a function td frequency and polarization.Lz = 0.75 cm; Lu = 0.75 cm; d = 0.07874 cm; e_ = 2.33; Loss tangent = 0.001; (oi,_ i) = (60 °, 180°).
6.25 GHz
Jr, '30 tA*im ' 15
0
12.25 GHz
Jr, .15Aim " - "_'__:::;
- P'_ 3" -
.3o k
0' ,< _--_
Figure 5. Surface current density d at the first and second resonances on the perfectly conducting patch described in figure 3.
case, but the general shape of the measured data isevident. Note that the peaks in the response have
decreased and broadened compared with the per-
fectly conducting patch. The complete set of scat-
tering results are shown in figure 7. The decreaseand broadening of the peaks is seen in each of the
responses. The surface resistance was then increasedto 11 _ over the entire patch. The measured and
calculated results for a00 are shown in figure 8. The
agreement between the two in this case is quite goodacross nlost of the frequency range with some dis-
agreement noted from 7.0 9.0 GHz. As expected,the resonant peaks in the response have decreased
and broadened in shape. The current distribution on
the patch is shown in figure 9. Although the gen-
eral shape of the current distribution is the same asfor the perfectly conducting patch, the amplitude has
been considerably reduced. The _r00 response for a
patch with a constant surface resistivity of 20 _2 is
shown in figure 10. The calculated and measured re-suits for this case agree across most of the frequency
band with only minor discrepancies at the lower fre-
quencies. This is thought to have been caused by the
measurement process, as evidenced by the rapid flue-tuations in the measured data at the lower frequen-
cies. With a 20 ft surface resistance on the patch, the
resonantpeaksseenin the previousresultsarenotevident.Theradarcrosssectionmaintainsa mono-tonicincreaseasfrequencyincreases.Thecalculatedresultsfor a00 for these four cases are summarizedin figure 11. As mentioned previously, as the sur-
face resistance increases the sharp resonant peaks in
the response gradually decrease and spread out. A
patch with a surface resistance of 20 fl has no no-
ticeable peaks in the scattering response. These re-
sults suggest that the addition of surface resistance tothe patch can be used to reduce the scattering from
the patch and could possibly be used to increase the
operating bandwidth of the antenna. However, this
increase in bandwidth may be at the expense of low-
ering the gain of the antenna.
-10
-20
-3O
-4O
-5O
N
12..
-- Calculated-- Measured
M
X
tt
-60 ' I , I , i _ I n I i I J I , I6 7 8 9 10 11 12 13 14
Frequency, GHz
Figure 6. Calculated and measured scattering from a rect-angular microstrip patch antenna with const'ant surfaceresistance of 5 9t. Lx = 0.75 cm; Ly = 0.75 cIn; d =0.07874 cm; er = 2.33; Loss tangent = 0.001; (0i,¢ i) =
(60 °, 180°).
Additional calculations have been performed oil
patches with surface resistance that varies as a func-tion of position on the patch surface. A patch that is
perfectly conducting but has a surface resistance of5 _ in each of the corners is shown in figure 12 along
with the measured and calculated results for aO0.
0-10-20-30
-40-50-60
d) -70.= -80
-90-100-110-120-130-140-150
6
CYoo
.... {Y¢¢,
...... rY,o (Yo,
, J L
7 8I i i i i
9 10 11 12 13 14
Frequency, GHz
Figure 7. Scattering from a rectangular microstrip patchwith a constant surface resistance of 5 gt as a fimc-
tion of frequency and polarization. Lz = 0.75 cm; L v =0.75 cm; d = 0.07874 cm; er = 2.33; Loss tangent = 0.001;(0 i, d)i) = (60 °, 180°).
-10
-2O
-3O
N
12.. M
-- Calculated--- Measured
-6C , I i I i I , l , i , I , I i l6 7 8 9 10 11 12 13 14
Frequency, GHz
Figure 8. Calculated and measured scattering from a rect-angular microstrip patch antenna with constant surfaceresistance of 11 FL L_ =0.75cm; L u=O.75cm; d=0.07874 era; er = 2.33; Loss tangent = 0.001; (0 i, Oi) =
(60 °, 180°).
Close agreement between the two is seen across the
entire frequency band, although the peaks in the
9
.050
Jx,
A/m .025
0
6.25 GHz12.25 GHz
.o5oJx, _ ......... ::_A/m .025 .......5-7
.050 .050
Jr, JY' .025A/m .025 A/m
0 0
Figure 9. Surface current density J at the first and second resonances on the resistive patch described in figure 8.
-10
-2O
-_ -30
N
12. M
Calculated- Measured
-50
_601tlllllJl*lJl_ll6 7 8 9 10 11 12 13
Frequency, GHz
I
14
-30&,_-40
oE-10
-20
-5O
-606
-- Perfect conductor
- Ill2 /N
.......
, I , I r I I , I I I I
7 8 9 10 11 12 13 14
Frequency, GHz
Figure 10. Calculated and measured scattering from a rect-
angular microstrip patch antenna with constant surface
Figure 11. Scattering from a rectangular microstrip patchantenna ms a function of frequency and surface resistance.Lx = 0.75 cm; Ly = 0.75 cm; d = 0.07874 cm; er = 2.33;Loss tangent = 0.001; (0i, O_) = (60 °, 180°).
10
0
-10
-2O
-3O
-40
-5O
-606
IIIIIIIIIIII:,2_
12..
_-- Lx --,q
III xIII
'.,9,d''
M
-- Calculated
Measured
_,-- __ _-_'- -
I i I l I l I J I i I _ I I I
7 8 9 10 11 12 13 14
Frequency, GHz
Figure 12. Calculated and mea.sured scattering from a rect-angular microstrip tmtch antenna with surface resis-tanceofR.s 5 _ for 0.375 cm < (Ix I and I,vl)< o.75 era.
.... i .... , .... i .... t .... L .... I .... _ ....
7 8 9 10 11 12 13 4
Frequency, GHz
Figure 13. Calculated results of scattering from a rect-angular microstrip patch antenna with surface re-sistance" R,_ for 0.375 cm < (Ix I and I,'/I) < 0.75 em.L_={).75cm; L v-0.75cm; d=(}.07874cm; e_ =2.33:Loss tangent = 0.001:(0 i, ¢i) -- (60 °, 180°).
measured response are not as high as was predicted.This discrepancy may have been caused by imper-
fections in the shape of the patch. In order to en-
sure that enough subdomains were used to model
the current for this patch, additional calculationswere performed with a higher number of subdonmins,
M = N = 16, and higher resistance on the patch cor-
ners. These results are shown in figure 13 along withresults for hi = N = 12. As the resistance on the
corners of the patch is increased, discontinuities in
the current density may result. To accurately model
this, a larger number of subdomains may" be neces-sary. However, little difference in the results is seen
in the figure with resistance in the corners as high
as 100 [_. The current distributions on the patch
described in figure 12 at the two resonant peaks are
shown in figure 14 and are similar to those shown pre-viously for patches with a constant surface resistance.
A similar patch and the accompanying c,00 results are
shown in figure 15. In this case the patch is perfectly
conducting in the center and has a 5 f_ surface re-
sistance around the perimeter. The resonant peaksin the response are lower than those of the previous
response, as is expected with the addition of resis-
tive material on a greater portion of the patch. A
microstrip patch with a 5 ft surface resistivity on the
patch edges is shown in figure 16. The resistance in
this case is on the patch edges that have the highercurrent density for the given excitation. A similar
patch is shown ill figure 17, but in this case the 5 f_
surface resistance is on the patch edges that have thelower current density for the given excitation. It is in-
teresting to note that the first resonance in figure 16
is considerably lower than the first resonance in fig-
ure 17, although the levels at the second resonancefor each case are nearly the same. A similar patch is
shown in figure 18, but the surface resistance on one
of the patch edges has been increased to 20 _. The
predicted shape of the response can be seen in themeasured data, although a frequency shift is clearly
evident. As expected, with the increase in the sur-
face resistance on the patch the level of the scattered
field decreases. As a final example, a patch that has
a 5 f_ surface resistance above the diagonal and isperfectly conducting below the diagonal has been an-
alyzed. This is shown in figure 19 along with the cal-
culated and measured radar cross section cro0. The
data show close agreement on the lower portion of
11
.30
Jx , .15A/m
0
.30
Jr, .15A/m
0
-10
-20
. -30
-40
-50
Figure 14. Surface current density J at the first and second resonances oi1 the patch described in figure 12.
. -f
12.. M
-- Calculated--- Measured
Y
N -iY
0 x
12.. M
-- Calculated
Measured
-50
-60 , i i I , i , I t I , i , i , I
6 7 8 9 10 I 1 12 13 14
Frequency, GHz
-10
-20
-3O
-40
-60 , I i I i I i I i I i I , i , I
6 7 8 9 10 11 12 13 14
Frequency, GHz
Figure 15. Calculated and measured scattering from a rect-
angular microstrip patch antenna with surface resis-
tance of Rs =5 fl for 0.375cm< (Ixl or lYl)<0.75cm.
Lx = 0.75 cm; Lu = 0.75 cm; d = 0.07874 cm; e_ = 2.33;
Loss tangent = 0.001; (0i, _i) = (60 o, 180o).
Figure 16. Calculated and measured scattering from a rect-
angular rnicrostrip patch antenna with surface re-
sistance of Rs = 5 f_ for 0.375 cm < lYl < 0.75 cm. L. =
0.75 cm; Ly = 0.75 em; d = 0.07874 cm; er = 2.33; Losstangent = 0.001; (0 i , _i ) = (60 ° ' 180°).
12
-- Calculated
- Measured
x
-606 7 8 9 10 11 12 13 14
Frequency, OHz
-10
-20
"_ -30
-40
-5O
-f• y
-60 i I i I i I t I i l i I t I , I6 7 8 9 10 11 12 13 14
Frequency, GHz
Figure 17. Calculated and measured scattering from a rect-
angular microstrip patch antenna with surface resistance
of R_ = 5 f_ for 0.375 cm < Ix I < (}.75 cm. Lz = 0.75 cm;
Ly = 0.75 cm; d - 0.{}787,i era; e r = 2.33; Loss tangent =
0.001; (Oi,O i) = (60 ° , 180°).
Figure 18. Calculated and measured scattering from a rect-
angular microstrip patch antenna with surface resis-
tance of Rs =512 for -0.75em<x<-0.375cm and
Rs =20f_ for 0.375em<x<(}.75cm. L_ =0.75cm;
Ly = 0.75 em; d = 0.07874 cm; er = 2.33; Loss tangent =
0.001; (0i,¢ i) = (60 ° , 180°).
N
6 7 8 9 10 11 12 13 14
Frequency. GHz
Figure 19. Calculated and measured scattering from a rectangular microstrip patch antenna with surface resistance of
R._ = 5 _ above the patch diagonal. Lz = 0.75 em Ly = 0.75 era; d = 0.07874 em er = 2.33; Loss tangent = 0.001;(0 i, ¢i) = (60 o, 180o).
30. Chang, Albert; and Mittra, Raj: Using Half-Plane So-
lutions in the Context of MM for Analyzing Large Flat
Structures With or Without Resistive Loading. IEEE
Trans. Antennas _ Propag., vol. AP-38, no. 7, July 1990,
pp. 1001-1009.
31. Shalaby, Abdel-Aziz T. K.: Spectral Domain Formula-
tion for Superconducting Microstrip Lines With Arbitrary
Strip Thickness. IEEE Antennas and Propagation Soci-
ety International Symposium 1992 Digest, Volume Two,
IEEE Catalog No. 92CH3178-1, IEEE Antennas and
Propagation Soc., 1992, pp. 990 993.
32. Lyons, W. G.; and Oates, D. E.: Microwave Characteriza-
tion of High-Tc Superconducting Thin Films and Devices.
IEEE Antennas and Propagation Society International
Symposium 1992 Digest, Volume Four, IEEE Catalog
No. 92CH3178-1, IEEE Antennas and Propagation Soc.,
1992, p. 2256.
15
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1. AGENCY USE ONLY(Leave blank) 2, REPORT DATE 3. REPORT TYPE AND DATES COVERED
August 1993 Technical Paper
4. TITLE AND SUBTITLE
Analysis of Microstrip Patch Antennas With Nonzero Surface
Resistance
6. AUTHOR(S)
David G. Shively and M. C. Bailey,
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
Joint Research Program Office NASA Langley Research Center
Electronics Integration Directorate Hampton, VA 23681-0001
Communications Electronics Command
Langley Research Center
Hampton, VA 23681-0001
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
U.S. Army Communications Electronics Command
Fort Monmouth, NJ 07703-5603and
National Aeronautics and Space Administration
Washington, DC 20546-0001
5. FUNDING NUMBERS
WU 505-64-20-54
P1L162211AH85
8. PERFORMING ORGANIZATION
REPORT NUMBER
L-17219
10. SPONSORING/MONITORING
AGENCY REPORT NUMBER
NASA TP-3362
CECOM TR-93-E-2
11. SUPPLEMENTARY NOTES
Shively: Joint Research Program Office, EID-CECOM, Langley Research Center, tlampton, VA;
Bailey: Langley Research Center, Hampton, VA.
12a. DISTRIBUTION/AVAILABILITY STATEMENT
Unclassified Ulflimited
Subject Category 32
12b. DISTRIBUTION CODE
13. ABSTRACT (Maximum 200 words)
The scattering properties of a microstrip patch antenna with nonzero surface impedance are examined. The
electric feld integral equation for a current element on a grounded dielectric slab is developed for a rectangulargeometry by using Galerkin's technique with subdomain pieeewise linear basis functions. The integral equation
includes a resistive boundary condition on the surface of the patch. The incident field on the patch is expressedas a function of incidence angle. The resulting system of equations is then solved for the unknown current
modes on the patch, and the radar cross section is calculated for a given scattering angle. Theoretical resultsin the form of radar cross section as a function of frequency are compared with results measured at the NASA